I'

The above example of the dynamic interactions occurring in the rough hole-boring process gives a flavor of the problem complexity, where the system has to be linearized first to be solved. This is because the physics of cutting processes, the metal cutting in particular, is a very complex issue, and a rigorous mathematical treatment of the plastic deformation in the cutting zone and the chip formation is still far from a satisfactory stage. Therefore, most attention has been paid to the static relationships between kinematic and geometrical parameters of the cutting process. However, it can be postulated the chatter irregularities of homogeneous materials are due to the nonlinearity of the process itself. To explore this view further, one starts with a steady state and considers a three-dimensional vector of the cutting force, fcut( ), which is dependent on the changes of the cutting parameters, a.i, the process constants, CJ and Pit and the Heaviside function, Hj,t(a.k). This can be written as

dissipative, and inertial properties of the machine tool structure, tool, and the workpiece are represented by a planar oscillator, which is excited by the cutting force components• fx and /y (see Fig. 3a). It is assumed that the relationship between the cutting forces and the chip geometry, namely the cutting process characteristics, is captured by orthogonal machining, where the cutting edge is parallel to the workpiece and normal to the cutting direction, as depicted in Fig. 3b. In our case, the cutting parameters, a.j should be understood as the depth of the cut, h, and the relative velocity, v,. Due to the vibration in the x direction, the relative velocity v, can cross the zero-value point; therefore, static and dynamic friction occurs. Thus, the cutting process characteristics as a function of the relative velocity cannot be expressed directly by formula (9); therefore, one can postulate the following relationships:

(I2) where two unknown functions, Kx(v,) and Kx(v,), need to be given explicitly. Becausefx and/1 are mutually related, one can be expressed by the another. This approach was adopted from the work of Hastings et al. [28], where the cutting forces for a wide class of technical materials are provided by

/y(y, X, y') = X(V,, Vf, h}fx(y, X, y') where

(I4)

where it was assumed that the force/y is mainly due to the friction x( ) acting on the rake surface. The friction velocity Vf is reduced due to shear plastic deformation, R, which is represented by the shear angle t/J (see Fig. 3b):

R = ctg(</J) (15) The cutting process starts with an initial depth of cut, h0 , where layers are taken out from the workpiece with the constant velocity v0 • The rest of the cutting parameters Ct - C4 and qo is fixed. Summarizing, the nonlinear

relationship between the cutting force fx and chip velocity is graphically presented in Fig. 3c, where for v, < 0 the excitation force is equal to zero. In reality, this force never disappears, as there is always a considerable friction force due to the compression force in the vertical spring. To make this approach more realistic, a Coulomb friction force acting in the x direction for the v, < 0 cases needs to be added. On the other hand, Eq. (6) should still be valid to predict the total force fx for the v, ~ 0 cases. A modified formula, which satisfies the above-listed conditions, is written as follows and is presented graphically in Fig. 3d:

fx(y. x,y') = qoh{H(v,)-1 - 1

- + sgn(v,)-1 Po }(ct(v,- 1) 2 + l)H(h)

where J4J is a static friction coefficient. The motion of the analyzed system can be described by a set of two

second-order differential equations, which are presented here in a nondimensional form:

where

(17)

(18)

Although one cannot deny the usefulness and elegance of approximate solutions, in this particular case, a more extensive analysis cannot be accomplished without numerical simulation. Therefore, Eqs. (17) and (18) are transformed to the system of four first-order differential equations, which can be written as

where

fx2(X2, X3, X4) =fx(y, X,y') fx4(X2, X3, X4) = /y(y, X, y')

DiscO

The expressions for the cutting forces [Eqs. (14) and (16)] have five different discontinuities, which can be labeled as one of two groups; either to the continuous discontinuity (unsmooth function), DiscC, or discontinuous discontinuity, DiscO. This classification was used to design a precise integration scheme, which is based on the fourth-order Runge-Kutta algorithm. The discontinuity DiscC is a product of a linear function and Heaviside function, whereas DiscO is a straightforward sign function (see Fig. 4).